Question

Find parametric equations for the line of intersection of the planes x y z=1 and x 2y 2z=1.

Answer 1

The parametric equations for the line of intersection of the planes x + y + z = 1 and x + 2y + 2z = 1 are x = t, y = (1 - t)/2, and z = (1 - t)/2.

Step-by-step explanation:

To find the parametric equations for the line of intersection of the planes x + y + z = 1 and x + 2y + 2z = 1, we can set up a system of equations using the given planes. The system of equations will have two variables, t and s, which represent the parameters for the line. We can solve for x, y, and z in terms of t and s, using the method of substitution. This will give us the parametric equations for the line of intersection.

1. Start by setting up the system of equations: x + y + z = 1 and x + 2y + 2z = 1
2. Since we are looking for parametric equations, we can express x, y, and z in terms of t and s. Let x = t and y = s.
3. Substitute these values into the equations and solve for z: t + s + z = 1 (equation 1) and t + 2s + 2z = 1 (equation 2)
4. From equation 1, we can solve for z: z = 1 - t - s
5. Substitute this value back into equation 2: t + 2s + 2(1 - t - s) = 1
6. Simplify the equation: 2 - t - 2s = 1
7. Rearrange the equation to solve for s: s = (1 - t)/2
8. Now, substitute this value back into equation 1 to solve for z: z = 1 - t - (1 - t)/2
9. Simplify the equation: z = (1 - t)/2
10. Therefore, the parametric equations for the line of intersection are: x = t, y = (1 - t)/2, and z = (1 - t)/2.